You need the open source mesa drivers for this and it does work as intended there. It won't work with any of the prop. drivers, though.

There's WINEDEBUG=+fps for games run under wine, although it isn't an overlay but rather it outputs on the terminal if you launch from there. Maybe hide all other output with -all if the games have a lot of errors/fixmes by default.

Lastly, some games have built in fps counters which are usually enabled by a config option.

[EDIT] I got it working and it does work on non-steam games too (although it flickers a lot, but is still readable), even though the guide says just typing "make" will build the 32bit version it doesn't, you have to make sure it's built too with "make voglperf32".
and if you want to use it with non-steam games you have to launch the game and manually supply the command line variables for VOGLPERF_CMD_LINE and LD_PRELOAD[/EDIT]